19 July 2018 || LoopFest XVII, Michigan State University

Rudi Rahnbased on work with Guido Bell, Bahman Dehnadi, Tobias Mohrmann (Siegen) and Jim Talbert (DESY)

Automated calculations of two-loop soft functions inSoft-Collinear Effective Theory

Outline

2

1. Resummation in SCET(a) Large logarithms(b) SCET and Factorisation(c) Renormalisation group resummation

2. Universal soft functions - dijet(d) Generic statements(e) NLO application(f) NNLO application(g) SoftSERVE

3. N-jet soft functions

Perturbative calculations involving widely separated scales exhibit large Sudakov logarithms spoiling convergence

Examples:‣ Event shapes‣ pT spectra‣ …

Origin: soft and collinear radiation, constrained real emission

Analytic resummation:‣ “direct QCD”: coherent branching‣ “Factorisation”: RG evolution

Large logarithms

3

↵ns ln2n

✓m2

Z

p2T

◆

↵ns ln2n ⌧

(Thrust)

(Z @ small pT)

[Catani, Trentadue, Turnock, Webber, ‘93]

[Collins, Soper, Sterman, ’85]SCET

SCET: Introduction

4

[Bauer, Fleming, Pirjol, Stewart, ’00] [Beneke, Chapovsky, Diehl, Feldmann, 02]

SCET is an effective theory for soft and collinear QCD field modes

It is highly observable dependent (field modes, relative scaling,…)

Formalises the expansion of QCD near the problematic limit

Allows us to derive factorisation theorems

5

SCET: FactorisationBrief Article

The Author

March 4, 2015

1

�0

d�

d⌧= �(⌧ )+

↵sCF

4⇡[(�2+

2⇡2

3)�(⌧ )�6[

1

⌧]+

�8[ln ⌧

⌧]+

] (1)

thrust T = maxn⌃i|pi · n|⌃i|pi|

(2)

1

�0

d�

d⌧= H(Q2, µ)

Zdp2L

Zdp2R J(p2L, µ) J(p2R, µ) S(⌧Q� p2L + p2R

Q,µ)

1

Thrust in SCET

In the two-jet limit ⌧ ! 0 the thrust distribution factorises as [Fleming, Hoang, Mantry,Stewart 07; Schwartz 07]

I 1�0

d�d⌧

= H(Q2, µ)

Zdp2

L

Zdp2

R J(p2L , µ) J(p2

R , µ) S⇣⌧Q �

p2L + p2

RQ

, µ⌘

multi-scale problem: Q2 � p2L ⇠ p2

R ⇠ ⌧Q2 � ⌧2Q2

hard collinear soft

H(Q2) J(p2R)J(p2L)

S(µ2S)

Hard function:

I on-shell vector form factor of a massless quark

H(Q2) =

2

I known to three-loop accuracy [Baikov, Chetyrkin, Smirnov, Smirnov, Steinhauser 09;Gehrmann, Glover, Huber, Ikizlerli, Studerus 10]

I also enters Drell-Yan and DIS in the endpoint region

Jet function:

I imaginary part of quark propagator in light-cone gauge

I J(p2) ⇠ ImhF.T.

D0��� n/n̄/

4 W†(0) (0) ̄(x)W (x) n̄/n/4

���0Ei

W (x) = P exp

igs

Z 0

�1ds n̄ · A(x + sn̄)

!

I known to two-loop accuracy (anomalous dimension to three-loop) [Becher, Neubert 06]

I also enters inclusive B decays and DIS in the endpoint region

Soft function:

I matrix element of Wilson lines along the directions of energetic quarks

I S(!) =X

X

���D

X���S†

n (0) Sn̄(0)���0E���

2�(! � n · pXn � n̄ · pXn̄

) Sn(x) = P exp

igs

Z 0

�1ds n · As(x + sn)

!

I known to two-loop accuracy (anomalous dimension to three-loop)[Kelley, Schwartz, Schabinger, Zhu 11; Monni, Gehrmann,

Luisoni 11; Hornig, Lee, Stewart, Walsh Zuberi 11]

J E T B R O A D E N I N G I N E F F E C T I V E F I E L D T H E O R Y G U I D O B E L LPA R T I C L E P H Y S I C S S E M I N A R – V I E N N A A P R I L 2 0 1 3

If the observable is compatible, SCET matrix elements factorise:

(for dijet thrust)

⌧ = 1� T ↵ns ln

2n ⌧

|CV |2 ⌃X|h0|Onn̄|Xi|2 (7)

= |CV |2 h0|h⇣̄0n̄W

0,†n̄

i h⇣̄0n̄W

0,†n̄

i† |0i h0| ⇥W 0n⇣

0n

⇤ ⇥W 0

n⇣0n

⇤† |0i h0|hS†n̄Sn

i hS†n̄Sn

i† |0i (8)

(9)

(1) ̄ �µ ! CV ? ⇣̄n̄ �µ? ⇣n (10)

(2) ⇣̄n̄ �µ? ⇣n ! ⇣̄n̄ S†n̄ �µ? Sn ⇣n (11)

(1) ̄(x) �µ (x) !Z

dsdt CV (s, t) ⇣̄n̄(x+ sn) �µ? ⇣n(x+ tn̄) (12)

(2) ̄(x) �µ (x) !Z

dsdt CV (s, t) ⇣̄0n̄ W 0,†

n̄ S†n̄(x�) �

µ? W 0

nSn(x+) ⇣0n (13)

µh ⇠ Q µj ⇠ Qp⌧ µs ⇠ Q⌧ (14)

H(Q2, µ) = H(Q2, µh)Uh(µh, µ) (15)

S(!) ' �(!) +↵s

4⇡S1(!) + ... with |A|2 = 64⇡2

k+k�CF (16)

! = |k+|1�a2 |k�|a2 (17)

p� ! pTpy

p+ ! pTpy (18)

M̄(⌧, k, l) = exp{�⌧ pT F (a, b, y)} (19)

2

SCET: Resummation

Functions in factorisation theorem only know one scale each e.g. for Thrust:

On their own they also require regularisation

Resummation in SCET:‣ Renormalise‣ Run from the natural scales to a common scale‣ RG running exponentiates logarithms

We need anomalous dimensions and renormalised functions

Added difficulty: SCET2 type observables show rapidity divergences➡ Additional analytic regulator required➡ Resummation via collinear anomaly

or rapidity renormalisation group6

µh = Q, µc =p⌧Q, µs = ⌧Q

<latexit sha1_base64="iTnZ2hlq95p1pfnPSGE/bgQjsJc=">AAACFnicbZDLSsNAFIYn9VbrLerSzWARXNSSiKCbQtGNyxbsBZoQJtNpO3RyceZEKKFP4cZXceNCEbfizrdx0mahrT8M/HznHM6c348FV2BZ30ZhZXVtfaO4Wdra3tndM/cP2ipKJGUtGolIdn2imOAhawEHwbqxZCTwBev445us3nlgUvEovINJzNyADEM+4JSARp555gSJN6o1K04lc7TmqHsJqQMkmeKM4gyrWgZw0zPLVtWaCS8bOzdllKvhmV9OP6JJwEKggijVs60Y3JRI4FSwaclJFIsJHZMh62kbkoApN52dNcUnmvTxIJL6hYBn9PdESgKlJoGvOwMCI7VYy+B/tV4Cgys35WGcAAvpfNEgERginGWE+1wyCmKiDaGS679iOiKSUNBJlnQI9uLJy6Z9XrWtqt28KNev8ziK6Agdo1Nko0tUR7eogVqIokf0jF7Rm/FkvBjvxse8tWDkM4foj4zPH5kcnl4=</latexit><latexit sha1_base64="iTnZ2hlq95p1pfnPSGE/bgQjsJc=">AAACFnicbZDLSsNAFIYn9VbrLerSzWARXNSSiKCbQtGNyxbsBZoQJtNpO3RyceZEKKFP4cZXceNCEbfizrdx0mahrT8M/HznHM6c348FV2BZ30ZhZXVtfaO4Wdra3tndM/cP2ipKJGUtGolIdn2imOAhawEHwbqxZCTwBev445us3nlgUvEovINJzNyADEM+4JSARp555gSJN6o1K04lc7TmqHsJqQMkmeKM4gyrWgZw0zPLVtWaCS8bOzdllKvhmV9OP6JJwEKggijVs60Y3JRI4FSwaclJFIsJHZMh62kbkoApN52dNcUnmvTxIJL6hYBn9PdESgKlJoGvOwMCI7VYy+B/tV4Cgys35WGcAAvpfNEgERginGWE+1wyCmKiDaGS679iOiKSUNBJlnQI9uLJy6Z9XrWtqt28KNev8ziK6Agdo1Nko0tUR7eogVqIokf0jF7Rm/FkvBjvxse8tWDkM4foj4zPH5kcnl4=</latexit><latexit sha1_base64="iTnZ2hlq95p1pfnPSGE/bgQjsJc=">AAACFnicbZDLSsNAFIYn9VbrLerSzWARXNSSiKCbQtGNyxbsBZoQJtNpO3RyceZEKKFP4cZXceNCEbfizrdx0mahrT8M/HznHM6c348FV2BZ30ZhZXVtfaO4Wdra3tndM/cP2ipKJGUtGolIdn2imOAhawEHwbqxZCTwBev445us3nlgUvEovINJzNyADEM+4JSARp555gSJN6o1K04lc7TmqHsJqQMkmeKM4gyrWgZw0zPLVtWaCS8bOzdllKvhmV9OP6JJwEKggijVs60Y3JRI4FSwaclJFIsJHZMh62kbkoApN52dNcUnmvTxIJL6hYBn9PdESgKlJoGvOwMCI7VYy+B/tV4Cgys35WGcAAvpfNEgERginGWE+1wyCmKiDaGS679iOiKSUNBJlnQI9uLJy6Z9XrWtqt28KNev8ziK6Agdo1Nko0tUR7eogVqIokf0jF7Rm/FkvBjvxse8tWDkM4foj4zPH5kcnl4=</latexit><latexit sha1_base64="iTnZ2hlq95p1pfnPSGE/bgQjsJc=">AAACFnicbZDLSsNAFIYn9VbrLerSzWARXNSSiKCbQtGNyxbsBZoQJtNpO3RyceZEKKFP4cZXceNCEbfizrdx0mahrT8M/HznHM6c348FV2BZ30ZhZXVtfaO4Wdra3tndM/cP2ipKJGUtGolIdn2imOAhawEHwbqxZCTwBev445us3nlgUvEovINJzNyADEM+4JSARp555gSJN6o1K04lc7TmqHsJqQMkmeKM4gyrWgZw0zPLVtWaCS8bOzdllKvhmV9OP6JJwEKggijVs60Y3JRI4FSwaclJFIsJHZMh62kbkoApN52dNcUnmvTxIJL6hYBn9PdESgKlJoGvOwMCI7VYy+B/tV4Cgys35WGcAAvpfNEgERginGWE+1wyCmKiDaGS679iOiKSUNBJlnQI9uLJy6Z9XrWtqt28KNev8ziK6Agdo1Nko0tUR7eogVqIokf0jF7Rm/FkvBjvxse8tWDkM4foj4zPH5kcnl4=</latexit>

[Becher, Bell, ‘11] [Becher, Neubert, ‘10]

[Chiu, Jain, Neill, Rothstein, ‘11]

Motivation for Automation

7

For NNLL resummation, some two-loop ingredients are required, we look at soft functions

Thrust[Becher, Schwartz, '08]

C-Parameter[Hoang, Kolodrubetz, Mateu, Stewart, '14]

Angularities[Bell, Hornig, Lee, Talbert, WIP]

…

Threshold Drell-Yan[Becher, Neubert, Xu, ’07]

W/Z/H @ large pT[Becher, Bell, Lorentzen, Marti, '13,'14]

Jet veto[Becher et al. ’13, Stewart et al., ’13]

…

So far we proceed observable by observable individually, e.g.:

Can this be made more systematic? It’s possible in full QCD…

Automation in QCD: CAESAR/ARES: [Banfi, Salam, Zanderighi, ’04], [Banfi, McAslan, Monni, Zanderighi, ’14]

Universal dijet soft functions

8

The generic form of the dijet soft function we get from the factorisation:

The measurement function (M) is observable dependent and harmless, e.g.

The matrix element is independent of the observable and is the source of divergences

Idea: isolate singularities at each order and calculate the associated coefficient numerically:

S(!) ' �(!) +↵s

4⇡S1(!) + ... with |A|2 = 64⇡2

k+k�CF (6)

! = |k+|1�a2 |k�|

a2 (7)

a ! 0 : thrust

a ! 1 :⇠ broadening

(8)

S({!i}, µ) = ⌃X,reg.

M({!i}, {kj})|hX|S†n(0)Sn̄(0)|0i|2 (9)

Sn(x) = Pexp(igs

Z 0

�1n ·As(x+ sn)ds) (10)

2

S(!) ' �(!) +↵s

4⇡S1

(!) + ... with |A|2 = 64⇡2

k+k�CF (28)

! = |k+|1�a2 |k�|a2 (29)

p� ! pTpy

p+ ! pTpy (30)

¯M(⌧, k, l) = exp{�⌧ pT F (a, b, y)} (31)

F (a, b, y) = yN/2ˆF (a, b, y) (32)

a ! 0 : thrust

a ! 1 :⇠ broadening

(33)

a ! 0 : thrust

a ! 1 :⇠ broadening

(34)

¯S2RR(⌧) = (µ2⌧2e�)2✏

8↵2sCFTFnf

⇡2

�(�4✏)p⇡�(1� ✏)�(1/2� ✏)

⇥Z 1

0dy

Z 1

0da

Z 1

0db

Z 1

0dt y�1+2N✏a1�2✏b�2✏

(a+ b)�2+2✏(1 + ab)�2+2✏

(4t¯t)�1/2�✏

⇥ N (a, b, t)

[(1� a)2 + 4at]2{[ ˆF (a, b, y)]4✏ + [

ˆF (a, b, 1/y]4✏}

S(!, µ) = ⌃

X,reg.M(!, {ki})|hX|S†

n(0)Sn̄(0)|0i|2 (35)

Sn(x) = Pexp(igs

Z 0

�1n ·As(x+ sn)ds) (36)

mY

i=1

=

Z(

⌫

n · ki )↵m �(k2i )✓(k

0i )d

dki (37)

¯S(1)(⌧, µ) =

µ2✏

(2⇡)d�1

Z�(k2) ✓(k0) R↵(⌫; k+, k�)

✓16⇡↵sCF

k+k�

◆¯M(⌧, k) ddk (38)

4

LQCD =

¯

i /D ) Lcollinear =¯⇣

✓in ·D + i /D?

1

in̄ ·Di /D?

◆/̄n2

⇣

Wc = Pexp

✓ig

Z 0

�1dsn̄ ·Ac(x+ sn̄)

◆(46)

⇣n(x) = Sn(x�)⇣0n(x) (47)

H(Q2, µ) = exp

4⇡�0�20

1

↵s(Q)

✓1� 1

r� ln r

◆�= 1��0

2

↵s(Q)

4⇡ln

2

✓Q2

µ2

◆+O(↵2

s), r =

↵s(µ)

↵s(Q)

ln

µ2

Q2, ln

µ2

⌧Q2, ln

µ2

⌧2Q2(48)

pµ ⇠ Q(1,�2,�)+,�,?

qµ ⇠ Q(�2, 1,�)

kµ ⇠ Q(�2,�2,�2)

(49)

¯S(⌧) ⇠ 1 + ↵s{c2✏2

+

c1✏1

+ c0} +O(↵2s) (50)

nµ= (1, 0, 0, 1) n̄µ

= (1, 0, 0,�1) (51)

n · n = 0 = n̄ · n̄ n · n̄ = 2 (52)

↵ns ln

2n

✓m2

H

p2T

◆(53)

I =

Z 1

0dx

Z 1

0dy(x+ y)�2+✏

(54)

I = I1 + I2 =

Z 1

0dx

Z x

0dy(x+ y)�2+✏

+

Z 1

0dy

Z y

0dx(x+ y)�2+✏

(55)

¯S(2)RR(⌧) =

µ4✏

(2⇡)2d�2

Zddk �(k2) ✓(k0)

Zddl �(l2) ✓(l0) |A(k, l)|2 ¯M(⌧, k, l) (56)

6

S(⌧, µ) =1

Nc

X

X

M(⌧, ki) Tr |h0|S†n̄(0)Sn(0)|Xi|2

Mthrust(⌧, {ki}) = exp

��⌧

Pi min(k+i , k

�i )

�(in Laplace space)

9

Universal soft functions: NLO

The virtual corrections are scaleless in dim reg, so the NLO soft function is:

To disentangle the soft and collinear divergences we parametrise suitably:

S(!) ' �(!) +↵s

4⇡S1(!) + ... with |A|2 = 64⇡2

k+k�CF (6)

! = |k+|1�a2 |k�|

a2 (7)

a ! 0 : thrust

a ! 1 :⇠ broadening

(8)

S({!i}, µ) = ⌃X,reg.

M({!i}, {kj})|hX|S†n(0)Sn̄(0)|0i|2 (9)

Sn(x) = Pexp(igs

Z 0

�1n ·As(x+ sn)ds) (10)

mY

i=1

=

Z(

⌫

n · ki)

↵m �(k2i )✓(k

0i )d

dki (11)

S̄(1)({⌧i}, µ) =µ2✏

(2⇡)d�1

Z�(k2) ✓(k0) R↵(⌫; k+) |A(k)|2 M̄({⌧i}, k+, k�, kT , {✓j}) ddk

(12)

k� ! kTpy

k+ ! kTpy (13)

2

We also must specify the measurement function M, and assume its form:

M(1)(⌧, k) = exp

��⌧ pT y

n2 f (y,#)

�kT

S(1)(⌧, µ) =µ2"

(2⇡)d�1

Z�(k2) ✓(k0)

16⇡↵sCF

k+k�M(⌧, k) ddk

S(1)(⌧, µ) ⇠ �(�2✏)

Z ⇡

0d#

Z 1

0dy y�1+n✏f(y,#)2✏

The kT integration can then be performed analytically, and yields the master formula:

10

Measurement functions: NLO examples

• For transverse thrust, , with beam axis, thrust axis s = sin ✓B , c = cos ✓B ✓B = \

M(1)(⌧, k) = exp

��⌧ pT y

n2 f (y,#)

�

Observable n f(y,#)

Thrust 1 1

Angularities 1�A 1

Recoil-free broadening 0 1/2

C-Parameter 1 1/(1 + y)

Threshold Drell-Yan �1 1 + y

W @ large pT �1 1 + y � 2

py cos ✓

e+e� transverse thrust 1

1spy

✓r(c cos ✓ +

⇣1py �p

y⌘

s2

⌘2+ 1� cos

2 ✓ ����c cos ✓ +

⇣1py �p

y⌘

s2

���◆

kT

Assume: Exponential function, motivated by Laplace space

Assume: is linear in mass dimension

Classify: How does the observable behave as y vanishes?

Assume: f positive and non-vanishing over almost all of phase space

This is enough to ensure the behaviour of the observable is under control in the critical limits:

Assumptions and classification: NLO

11

exp(�⌧!({ki})) =

Z 1

0d! exp(�⌧!) �(! � !({ki}))

M = exp(�⌧kT ˆf(y,#))

!

M = exp(�⌧kT yn2 f(y,#))

(kT ! 0) )

(y ! 0) )SoftCollinear

vanishes, fixed by mass dimensionf finite

Universality: NLO vs. NNLO

12

Figure 3: Next-to-leading corrections to the soft function. The one-loop virtual diagrams arescaleless and vanish.

The Laplace and Fourier transforms can be performed explicitly, yielding

S(τL, τR, zL, zR) = 1 +CFαs

π

1

β − α

21−α−β−2ϵ Γ(−α− β − 2ϵ)

Γ(1− ϵ)

(µ2eγE

)ϵ(ν21Q

)α(ν22Q

)β

×[τα+β+2ϵL 2F1

(− α + β + 2ϵ

2,1− α− β − 2ϵ

2, 1− ϵ,−z2L

)− (L ↔ R)

].

(20)We now expand this expression by taking the limits β → 0, α → 0, and ϵ → 0. The orderin which the analytic regulators are taken to zero is arbitrary, but it is important that thelimit ϵ→ 0 is taken at the end, since only then the QCD result is independent of the analyticregularization. The final expression for the bare soft function is

S(τL, τR, zL, zR) = 1 +CFαs

4π

{

− 2

ϵ2− 2

ϵln(µ2τ̄ 2L)− ln2(µ2τ̄ 2L)

+ 4

(1

α+ ln

ν21 τ̄LQ

)[1

ϵ+ ln(µ2τ̄ 2L) + 2 ln

√1 + z2L + 1

4

]

+ 8Li2

(−

√1 + z2L − 1

√1 + z2L + 1

)+ 4 ln2

√1 + z2L + 1

4+

5π2

6− (L ↔ R)

}

,

(21)

where τ̄L = τLeγE . The coefficients of the 1/α poles are equal and opposite to those in the jetfunctions, see (18), so that these divergences cancel in the product J L J R S.

Let us now use the expressions derived above to compute the differential cross section atone-loop order. In this approximation we only need the convolutions of the tree-level softfunction with the one-loop jet functions and vice versa. Since the tree-level soft functioninvolves δ-functions in the transverse momenta, we only need the jet function JL(b, p⊥ = 0)given in (16). The resulting expression can be refactorized in the form

1

σ0

d2σ

dbL dbR= H(Q2)Σ(bL)Σ(bR) , (22)

where

Σ(b) = δ(b) +CFαs

4π

eϵγE

Γ(1− ϵ)

1

b

(µb

)2ϵ(4 ln

Q2

b2− 6− 2ϵ

). (23)

7

(a) (b) (c) (d)

(e) (f) (g) (h)

Figure 3: Diagrams that give non-vanishing contributions to the soft function at NNLO. Inaddition there are mirror-symmetrical graphs, which we take into account by multiplying eachdiagram with a symmetry factor si, where sa = sb = sf = sg = 2, sc = 1 and sd = se = sh = 4,see text.

4 Two-loop anomaly coefficient

4.1 Setup of the calculation

The anomaly coefficient FB(τ, z, µ) can be extracted from the divergences in the analyticregulator of any of the soft and jet functions. Here, we will consider the two-loop soft function,since in this case there are again up to two partons in the final state and the integrals aresimilar to the ones that we encountered in the calculation of the one-loop jet function. As thedivergences cancel in each hemisphere independently, it is sufficient to consider emissions intoone of the hemispheres only. To be specific, we focus on the emissions into the left hemisphere.

At two-loop order the purely virtual corrections are again scaleless and vanish. Amongthe mixed virtual-real and double real emissions, only the diagrams in Figure 3 give non-vanishing contributions. The same matrix elements also arise in other two-loop computationsof soft functions [24–32], what makes our case different are the phase-space constraints andthe necessity of working with an additional analytic regulator. In the following, we denote theindividual contributions of these diagrams to the soft function by

S(2i)L (bL, bR, p

⊥L , p

⊥R), i = a, b, . . . , h (42)

and similarly for the Laplace-Fourier transformed soft function S.The diagrams in Figure 3 have counterpart diagrams that follow by exchanging nµ ↔ n̄µ

as well as complex conjugation. It turns out that the matrix elements of diagrams (d), (e)and (g) are not symmetric under nµ ↔ n̄µ. In an integral over a symmetric phase space,

13

NLO:

NNLO:

13

Universality: NLO vs. NNLOConsider the double real emission:

I = I1 + I2 =

Z 1

0dx

Z x

0dy(x+ y)�2+✏

+

Z 1

0dy

Z y

0dx(x+ y)�2+✏

(25)

¯SRR(⌧) =µ4✏

(2⇡)2d�2

Zddk �(k2) ✓(k0)

Zddl �(l2) ✓(l0) |A(k, l)|2 ¯M(⌧, k, l) (26)

|A(k, l)|2 = 128⇡2↵2sCFTFnf

2k · l(k� + l�)(k+ + l+)� (k�l+ � k+l�)

(k� + l�)2(k+ + l+)2(2k · l)2 (27)

I1 =

Z 1

0dx

Z 1

0dy x�1+✏

(1 + t)�2+✏(28)

I2 =

Z 1

0dy

Z 1

0dx y�1+✏

(1 + t)�2+✏(29)

pµ =

n̄µ

2

n · p+ nµ

2

n̄ · p+ p?,µ ⌘ (p+, p�, p?) (30)

(31)

p2 = p+p� + p2? (32)

(33)

p · q =

1

2

p+ · q� +

1

2

p� · q+ + p? · q? (34)

4

The matrix elements are no longer nice and easy, see e.g., the CFTFnf color structure:

The singularities are partially overlapping, not as easy to extract, but it’s possibleWe then again assume the form of the measurement function:

pµ ⇠ Q(1,�2,�)+,�,?

qµ ⇠ Q(�2, 1,�)

kµ ⇠ Q(�2,�2,�2)

(48)

S ⇠ {c2✏2

+c1✏1

+ c0} (at order ↵1s) (49)

nµ = (1, 0, 0, 1) n̄µ = (1, 0, 0,�1) (50)

n · n = 0 = n̄ · n̄ n · n̄ = 2 (51)

↵ns ln

2n

✓m2

H

p2T

◆(52)

I =

Z 1

0dx

Z 1

0dy(x+ y)�2+✏ (53)

I = I1 + I2 =

Z 1

0dx

Z x

0dy(x+ y)�2+✏ +

Z 1

0dy

Z y

0dx(x+ y)�2+✏ (54)

S̄2RR(⌧) =

µ4✏

(2⇡)2d�2

Zddk �(k2) ✓(k0)

Zddl �(l2) ✓(l0) |A(k, l)|2 M̄(⌧, k, l) (55)

S̄2RR(⌧) = (⇡µ2e�)2✏

⌦d�2⌦d�3

1024⇡6

Z 1

0dk+

Z 1

0dk�

Z 1

0dl+

Z 1

0dl�

Z 1

�1d cos ✓(k+k�l+l�)

�✏ sind�5 ✓

⇥|A(k, l)|2 M̄(⌧, k, l)

|A(k, l)|2 = 128⇡2↵2sCFTFnf

2k · l(k� + l�)(k+ + l+)� (k�l+ � k+l�)

(k� + l�)2(k+ + l+)2(2k · l)2 (56)

Aµ(x) ! Aµc (x) +Aµ

s (x) µ(x) ! µc (x) +

µs (x) (57)

⇣(x) =/n/̄n

4 c(x), ⌘(x) =

/̄n/n

4 c(x) (58)

6

M(2)(⌧, k, l) = exp

��⌧ pT y

n2 F (y, a, b,#,#k,#l)

�

Why is this enough?

2-loop - Correlated emissions:

Matrix element divergent in four critical limits: Behaviour

14

Only one unconstrained variableVariable definition ensures commuting limits

(Global soft)

(Individual soft)

(emissions collinear)

(Global “jet-collinear”)

fixed by IR safety

fixed by collinear safety

fixed by mass dimension

unconstrained Classify!

M(2)(⌧, k, l) = exp

��⌧ pT y

n2 F (y, a, b,#,#k,#l)

�

CFCA, CFTfnf

2-loop - Uncorrelated emissions:

4 critical limits: Behaviour

15

Two “unconstrained” limitsWorse: Overlapping zeroes:

(Global soft)

(individual soft)

(one emission “jet-collinear”)( )2unconstrained

fixed by IR safety

fixed by mass dimension

!(k, l) = kT yn2k f(yk) + lT y

n2l f(yl)

Solution: adapt parametrisation for kT, lT:

kT = pTb

1 + b

✓ pyl

1 + yl

◆n

, lT = pT1

1 + b

✓ pyk

1 + yk

◆n

C2F

qTqT

2-loop - Uncorrelated emissions:

4 critical limits: Behaviour

16

Two “unconstrained” limitsWorse: Overlapping zeroes:

(Global soft)

(individual soft)

(one emission “jet-collinear”)( )2unconstrained

fixed by IR safety

fixed by mass dimension

!(k, l) = kT yn2k f(yk) + lT y

n2l f(yl)

Solution: adapt parametrisation for kT, lT:

kT = pTb

1 + b

✓ pyl

1 + yl

◆n

, lT = pT1

1 + b

✓ pyk

1 + yk

◆n

C2F

qTqT

2-loop - Uncorrelated emissions:

4 critical limits: Behaviour

17

Suitable parametrisation solves overlapping limit problem

(Global soft)

(individual soft)

(one emission “jet-collinear”)( )2unconstrained

fixed by IR safety

fixed by mass dimension

C2F

M(2)(⌧ ; k, l) = exp

⇣�⌧ qT y

n2k y

n2l G(yk, yl, b,#,#k,#l)

⌘

<latexit sha1_base64="1Qg2RNXclt8kTJT7n8QGZqixljg=">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</latexit><latexit sha1_base64="1Qg2RNXclt8kTJT7n8QGZqixljg=">AAACdnicbVFda9swFJW9ry77yranMRiiocwBL9hh0EIZlO1hexl00LSFKDWycp2IyLInXZca45+wP7e3/Y697HFyGmjX7oLQ0Tnn6uMoLZW0GEW/PP/O3Xv3H2w97D16/OTps/7zF8e2qIyAiShUYU5TbkFJDROUqOC0NMDzVMFJuvrU6SfnYKws9BHWJcxyvtAyk4Kjo5L+D5ZzXAqumq/tWROMh23AkFf7q1ANPzC4KJmCDIN3HUlZSL8nR91UJ6szlhkuGt0247ZO1PVl5/gcOE/ohDAN2Tk3uATkV8hpV1gNmZGLJQ6T/iAaReuit0G8AQOyqcOk/5PNC1HloFEobu00jkqcNW5jKRS0PVZZKLlY8QVMHdQ8Bztr1rG1dMcxc5oVxg2NdM1e72h4bm2dp87ZhWRvah35P21aYbY3a6QuKwQtLg/KKkWxoN0f0Lk0IFDVDnBhpLsrFUvu4kP3Uz0XQnzzybfB8XgUR6P42/vBwcdNHFvkNdkmAYnJLjkgX8ghmRBBfnuvvG1v4P3x3/g7/ttLq+9tel6Sf8qP/gJgNb46</latexit><latexit sha1_base64="1Qg2RNXclt8kTJT7n8QGZqixljg=">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</latexit><latexit sha1_base64="1Qg2RNXclt8kTJT7n8QGZqixljg=">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</latexit>

The completed frameworkSemi-Analytic expressions are available for anomalous dimensions

For finite parts we have an implementation using pySecDec and a dedicated C++ based program:

SoftSERVE uses the Cuba library’s Divonne integrator, implements numerical improvements, and has multiprecision variable support

Soon™ to be found on HEPForge, paper in preparation18

(Soft function Simulation and Evaluation for Real and Virtual Emissions)

[1805.12414]

[Borowka, Heinrich, Jahn, Jones, Kerner, Schlenk, Zirke, 1703.09692]

[Hahn, hep-ph/0404043]

SoftSERVE

First release will be correlated emissions onlyUser required input (C++ syntax):‣ correlated: two functions F (roughly: one for each hemisphere)‣ uncorrelated: three functions G‣ two parameters‣ optional: parameter values, integrator settings

Two integrator settings pre-definedScripts available for renormalisation in Laplace and momentum space, and for dealing with Fourier space observables

19

Deviations from analytic results compatible with 1 error estimate

A few results

Derived in few minutes to hours on an 8 core desktop machine

20

�

�1 = �CA1 CFCA + �

Nf

1 CFTFnf c2 = cCA2 CFCA + c

Nf

2 CFTFnf +1

2(c1)

2

Soft function �CA1 �

nf

1 cCA2 c

nf

2

Thrust

[Kelley et al, ’11]

[Monni et al, ’11]

15.7945(15.7945)

3.90981(3.90981)

�56.4992(�56.4990)

43.3902(43.3905)

C-Parameter

[Hoang et al, ’14]

15.7947(15.7945)

3.90980(3.90981)

�57.9754[�58.16± 0.26]

43.8179[43.74± 0.06]

Threshold Drell-Yan

[Belitsky, ’98]

15.7946(15.7945)

3.90982(3.90981)

6.81281(6.81287)

�10.6857(�10.6857)

W @ large pT[Becher et al, ’12]

15.7947(15.7945)

3.90981(3.90981)

�2.65034(�2.65010)

�25.3073(�25.3073)

Transverse Thrust

[Becher, Garcia, Piclum, ’15]

�158.278[�148±20

30]

19.3955[18±2

3]

parameterdependent

parameterdependent

Results: Angularities

21

Generalisation of thrustObeys non-abelian observationNew result, will feature in NNLL’ resummation paper soon™

c2CA

-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4

-100

-50

0

50

A

c2n f

-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.40

20

40

60

80

100

A

g1CA

-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.40

5

10

15

20

25

A

g1n f

-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4-1

0

1

2

3

4

5

6

A

Results: Soft drop jet mass

Jet grooming procedures remove radiation from jets to reveal substructureFor the soft drop groomer multiple observables have been proposed and factorised in [Frye et al, 1603.09338].For Cambridge/Aachen clustering and jet mass as the observable, EVENT2 fits were presented for the anomalous dimensionBreaks non-abelian exponentiation

22

γ1CA

SoftSERVEFrye et al.

0.0 0.5 1.0 1.5 2.0

-5

0

5

10

15

20

β

γ1CF

SoftSERVEFrye et al.

0.0 0.5 1.0 1.5 2.0

-30

-20

-10

0

10

β

γ1nf

SoftSERVEFrye et al.

0.0 0.5 1.0 1.5 2.0

-5

0

5

10

15

β

Extension to N jet directionsThere are now more jet/beam directions -> more Wilson lines:

23

‣ Tripole and Quadrupole diagrams✓ Assume non-abelian exponentiation:

only one tripole (RV)‣ Dipole directions are no longer back-to-back

✓ Use boost-invariant parametrisation✓ Consequence: transverse space can acquire

temporal direction‣ more complicated angular integrations

✓ 5 angles instead of 3 at NNLO‣ external geometry must be sampled

k = (k · na, k · nb, k?)

k = (k · n, k · n̄, k?)

S(⌧, µ) =X

X

M(⌧, ki) h0| (Sn1Sn2Sn3 ...)† |XihX|Sn1Sn2Sn3 ...|0i

Protonbeam bProton

beam a

Jet

0.0 0.2 0.4 0.6 0.8 1.0- 120- 100- 80- 60- 40- 20020

gg → g qq̄ → g qg → q

na1/ 2

1-Jettiness

24

Kinematics and Sampling

Finite terms for different channels (preliminary)

Dots: [Bell, Dehnadi, Mohrmann, RR, in preparation]Lines: [Campbell, Ellis, Mondini, Williams, ‘17]

na · nb = 2

na · n1 = 1� cos ✓ = na1

nb · n1 = 1 + cos ✓<latexit sha1_base64="r86ZtTQWeZEXVcvV5IqF2GYuO64=">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</latexit><latexit sha1_base64="r86ZtTQWeZEXVcvV5IqF2GYuO64=">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</latexit><latexit sha1_base64="r86ZtTQWeZEXVcvV5IqF2GYuO64=">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</latexit><latexit sha1_base64="r86ZtTQWeZEXVcvV5IqF2GYuO64=">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</latexit>

[Boughezal, Liu, Petriello, ’15;Campbell, Ellis, Mondini, Williams, ‘17]

2-Jettiness

25

Some preliminary results - dipoles

Proton beam bProton

beam a

Jet 1

Jet 2

0.0 0.2 0.4 0.6 0.8 1.0- 100

- 80

- 60

- 40

- 20

0

0.0 0.2 0.4 0.6 0.8 1.0050100150200250300

I nfab I CAab

na1/ 2 na1/ 2

Kinematics and Sampling

na · nb = n1 · n2 = 2

na · n1 = nb · n2 = 1� cos ✓ = na1

na · n2 = nb · n1 = 1 + cos ✓<latexit sha1_base64="WYRmEj0UC5VmgvQufSpV4Nqi8/k=">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</latexit><latexit sha1_base64="WYRmEj0UC5VmgvQufSpV4Nqi8/k=">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</latexit><latexit sha1_base64="WYRmEj0UC5VmgvQufSpV4Nqi8/k=">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</latexit><latexit sha1_base64="WYRmEj0UC5VmgvQufSpV4Nqi8/k=">AAACcnicbVHbSiMxGM6M66ketrrsjQsaLYqwWiZF2L0piN7spYJVoVOGf9LUBjPJkPwjlKEP4Ot5t0+xN/sAm3YHt1v9IfDlO5DkS5or6TCKfgbhwofFpeWV1dra+sbmx/rW9q0zheWiw40y9j4FJ5TUooMSlbjPrYAsVeIufbyc6HdPwjpp9A2OctHL4EHLgeSAnkrqzzoBGvO+QaqTlB61dcJe9y3apq04rs162NST/vO02WnMjaMxDgWCT+ikBDaei7XmYswb2dfZYFJvRM1oOvQtYBVokGqukvpL3De8yIRGrsC5Loty7JVgUXIlxrW4cCIH/ggPouuhhky4XjmtbEwPPdOnA2P90kin7GyihMy5UZZ6ZwY4dPPahHxP6xY4+N4rpc4LFJr/PWhQKIqGTvqnfWkFRzXyALiV/q6UD8ECR/9LNV8Cm3/yW3DbarKoya7PGucXVR0r5As5IMeEkW/knPwgV6RDOPkVfA52g73gd7gT7odVd2FQZT6R/yY8+QOugrZI</latexit>

2-Jettiness

26

Some preliminary results - tripole

Protonbeam bProton

beam a

Jet 1

Jet 2

Kinematics and Sampling

na · nb = n1 · n2 = 2

na · n1 = nb · n2 = 1� cos ✓ = na1

na · n2 = nb · n1 = 1 + cos ✓<latexit sha1_base64="WYRmEj0UC5VmgvQufSpV4Nqi8/k=">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</latexit><latexit sha1_base64="WYRmEj0UC5VmgvQufSpV4Nqi8/k=">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</latexit><latexit sha1_base64="WYRmEj0UC5VmgvQufSpV4Nqi8/k=">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</latexit><latexit sha1_base64="WYRmEj0UC5VmgvQufSpV4Nqi8/k=">AAACcnicbVHbSiMxGM6M66ketrrsjQsaLYqwWiZF2L0piN7spYJVoVOGf9LUBjPJkPwjlKEP4Ot5t0+xN/sAm3YHt1v9IfDlO5DkS5or6TCKfgbhwofFpeWV1dra+sbmx/rW9q0zheWiw40y9j4FJ5TUooMSlbjPrYAsVeIufbyc6HdPwjpp9A2OctHL4EHLgeSAnkrqzzoBGvO+QaqTlB61dcJe9y3apq04rs162NST/vO02WnMjaMxDgWCT+ikBDaei7XmYswb2dfZYFJvRM1oOvQtYBVokGqukvpL3De8yIRGrsC5Loty7JVgUXIlxrW4cCIH/ggPouuhhky4XjmtbEwPPdOnA2P90kin7GyihMy5UZZ6ZwY4dPPahHxP6xY4+N4rpc4LFJr/PWhQKIqGTvqnfWkFRzXyALiV/q6UD8ECR/9LNV8Cm3/yW3DbarKoya7PGucXVR0r5As5IMeEkW/knPwgV6RDOPkVfA52g73gd7gT7odVd2FQZT6R/yY8+QOugrZI</latexit>

0.0 0.2 0.4 0.6 0.8 1.0

- 4000

- 2000

0

2000

4000

na1/ 2

Conclusion

We have developed a framework to systematically compute generic NNLO dijet soft functions for wide ranges of observables at lepton and hadron colliders

27

An extension to N-jet observables seems possible, and we have already re-derived a few known results and are working on new ones

SCET provides an efficient, analytic approach to high-order resummations necessary for precision collider physics.

The program(s) based on this framework will soon™ be released into the wild

That’s all folks!

28

Thank you!

Parametrisation uncorrelated

29

k+ = qTb

1 + b

pyk

✓ pyl

1 + yl

◆n

k� = qTb

1 + b

1pyk

✓ pyl

1 + yl

◆n

l+ = qT1

1 + b

pyl

✓ pyk

1 + yk

◆n

l� = qT1

1 + b

1pyl

✓ pyk

1 + yk

◆n

yk =k+k�

b =

sk+k�l+l�

1+n✓l+ + l�k+ + k�

◆n

yl =l+l�

qT =pl+l�

k+ + k�p

k+k�

!n

+p

k+k�

l+ + l�p

l+l�

!n

The parametrisation for the uncorrelated emissions

leads to divergences in b, yk, yl, qT (analytic)

Parametrisation correlated

30

The parametrisation for the correlated emissions

leads to divergences in y, b, pT (analytic), and an overlapping divergence in a 1 (with transverse angle)

a =

sk�l+l�k+

=

rylyk

b =

sk+k�l+l�

=kTlT

y =k+ + l+k� + l�

pT =p(k+ + l+)(k� + l�)

k+ = pTb

a+ b

py k� = pT

ab

1 + ab

1py

l+ = pTa

a+ b

py l� = pT

1

1 + ab

1py